Kubota's cubic theta function 
is an automorphic
form on three-dimensional hyperbolic space,
represented by pairs 
with 
a complex number and 
. It is obtained, up to normalization, as the complex residue at a simple
pole of a cubic metaplectic Eisenstein
series over the imaginary quadratic field 
(Kubota 1969, Patterson 1977).
Let ![O=Z[omega]](/images/equations/KubotasCubicThetaFunction/Inline6.svg)
be the ring of Eisenstein
integers, where 
,
and set 
. At the cusp
at infinity, 
is periodic in 
under translations by 
.
Holding the height 
fixed gives a Fourier series
in the real part and imaginary
part of 
. In the normalization of Dunn and Radziwiłł
(2024), this expansion is
 |
where 
is the modified Bessel function
of the second kind of order 
and the overline denotes the complex
conjugate, so ![muz+muz^_=2Re[muz]](/images/equations/KubotasCubicThetaFunction/Inline16.svg)
.
The constant term is independent of 
.
For each nonzero frequency 
, the Fourier coefficient
has height dependence 
and arithmetic factor 
. Equivalently, the height factor is 
for 
,
where 
is the decaying Airy function (NIST DLMF).
For example, if 
is an Eisenstein prime
with 
, Patterson's coefficient formula gives
 |
where 
is the normalized cubic Gauss sum (Patterson 1977,
Dunn and Radziwiłł 2024). The transformation law of 
relates weighted sums of its Fourier
coefficients at different cusps. This
provides a way to estimate sums of cubic Gauss sums
and is used in the announced argument for the quasi-Riemann
hypothesis (OpenAI 2026).
-Functions." Oct. 5, 2026.